Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 34:c9ad0d97ce41
fix oridinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 22 May 2019 11:52:49 +0900 |
parents | 2b853472cb24 |
children | 4d64509067d0 |
rev | line source |
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16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
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2 module ordinal-definable where |
3 | 3 |
14
e11e95d5ddee
separete constructible set
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4 open import zf |
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5 open import ordinal |
3 | 6 |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
3 | 8 |
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9 open import Relation.Binary.PropositionalEquality |
3 | 10 |
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11 open import Data.Nat.Properties |
6 | 12 open import Data.Empty |
13 open import Relation.Nullary | |
14 | |
15 open import Relation.Binary | |
16 open import Relation.Binary.Core | |
17 | |
27 | 18 -- Ordinal Definable Set |
11 | 19 |
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20 record OD {n : Level} : Set (suc n) where |
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21 field |
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22 def : (x : Ordinal {n} ) → Set n |
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23 |
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24 open OD |
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25 open import Data.Unit |
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26 |
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27 postulate |
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28 od→ord : {n : Level} → OD {n} → Ordinal {n} |
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29 |
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30 ord→od : {n : Level} → Ordinal {n} → OD {n} |
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31 ord→od x = record { def = λ y → x ≡ y } |
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32 |
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33 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
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34 _∋_ {n} a x = def a ( od→ord x ) |
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35 |
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36 _c<_ : { n : Level } → ( a x : OD {n} ) → Set n |
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37 x c< a = a ∋ x |
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38 |
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39 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
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40 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
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41 |
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42 postulate |
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43 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord x |
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44 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od x |
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45 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x |
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46 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
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47 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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48 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ |
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49 |
28 | 50 HOD = OD |
51 | |
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52 od∅ : {n : Level} → HOD {n} |
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53 od∅ {n} = record { def = λ _ → Lift n ⊥ } |
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54 |
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55 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) |
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56 ∅1 {n} x (lift ()) |
28 | 57 |
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58 ∅3 : {n : Level} → ( x : Ordinal {n}) → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
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59 ∅3 {n} x = TransFinite {n} c1 c2 c3 x where |
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60 c0 : Nat → Ordinal {n} → Set n |
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61 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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62 c1 : ∀ (lx : Nat ) → c0 lx (record { lv = Suc lx ; ord = ℵ lx } ) |
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63 c1 lx not with not ( record { lv = lx ; ord = Φ lx } ) |
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64 ... | t with t (case1 ≤-refl ) |
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65 c1 lx not | t | () |
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66 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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67 c2 Zero not = refl |
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68 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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69 ... | t with t (case1 ≤-refl ) |
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70 c2 (Suc lx) not | t | () |
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71 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) |
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72 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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73 ... | t with t (case2 Φ< ) |
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74 c3 lx (Φ .lx) d not | t | () |
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75 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 76 ... | t with t (case2 (s< s<refl ) ) |
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77 c3 lx (OSuc .lx x₁) d not | t | () |
34 | 78 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } ) |
79 ... | t with t (case2 (s< (ℵΦ< {_} {_} {Φ (Suc lx)}))) | |
80 c3 .(Suc lx) (ℵ lx) d not | t | () | |
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81 |
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82 ∅2 : {n : Level} → od→ord ( od∅ {n} ) ≡ o∅ {n} |
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83 ∅2 {n} = {!!} |
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84 |
28 | 85 HOD→ZF : {n : Level} → ZF {suc n} {suc n} |
86 HOD→ZF {n} = record { | |
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87 ZFSet = OD {n} |
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88 ; _∋_ = λ a x → Lift (suc n) ( a ∋ x ) |
28 | 89 ; _≈_ = _≡_ |
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90 ; ∅ = od∅ |
28 | 91 ; _,_ = _,_ |
92 ; Union = Union | |
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93 ; Power = Power |
28 | 94 ; Select = Select |
95 ; Replace = Replace | |
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96 ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } } |
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97 ; isZF = isZF |
28 | 98 } where |
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99 Replace : OD {n} → (OD {n} → OD {n} ) → OD {n} |
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100 Replace X ψ = sup-od ψ |
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101 Select : OD {n} → (OD {n} → Set (suc n) ) → OD {n} |
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102 Select X ψ = record { def = λ x → select ( ord→od x ) } where |
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103 select : OD {n} → Set n |
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104 select x with ψ x |
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105 ... | t = Lift n ⊤ |
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106 _,_ : OD {n} → OD {n} → OD {n} |
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107 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } |
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108 Union : OD {n} → OD {n} |
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109 Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y } |
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110 Power : OD {n} → OD {n} |
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111 Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) } |
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112 ZFSet = OD {n} |
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113 _∈_ : ( A B : ZFSet ) → Set n |
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114 A ∈ B = B ∋ A |
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115 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n |
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116 _⊆_ A B {x} = A ∋ x → B ∋ x |
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117 _∩_ : ( A B : ZFSet ) → ZFSet |
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118 A ∩ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∧ (Lift (suc n) ( B ∋ x ) )) |
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119 _∪_ : ( A B : ZFSet ) → ZFSet |
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120 A ∪ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∨ (Lift (suc n) ( B ∋ x ) )) |
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121 infixr 200 _∈_ |
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122 infixr 230 _∩_ _∪_ |
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123 infixr 220 _⊆_ |
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124 isZF : IsZF (OD {n}) (λ a x → Lift (suc n) ( a ∋ x )) _≡_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) |
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125 isZF = record { |
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126 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } |
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127 ; pair = pair |
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128 ; union→ = {!!} |
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129 ; union← = {!!} |
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130 ; empty = empty |
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131 ; power→ = {!!} |
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132 ; power← = {!!} |
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133 ; extentionality = {!!} |
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134 ; minimul = minimul |
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135 ; regularity = {!!} |
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136 ; infinity∅ = {!!} |
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137 ; infinity = {!!} |
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138 ; selection = {!!} |
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139 ; replacement = {!!} |
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140 } where |
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141 open _∧_ |
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142 pair : (A B : OD {n} ) → Lift (suc n) ((A , B) ∋ A) ∧ Lift (suc n) ((A , B) ∋ B) |
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143 proj1 (pair A B ) = lift ( case1 refl ) |
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144 proj2 (pair A B ) = lift ( case2 refl ) |
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145 empty : (x : OD {n} ) → ¬ Lift (suc n) (od∅ ∋ x) |
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146 empty x (lift (lift ())) |
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147 union→ : (X x y : OD {n} ) → Lift (suc n) (X ∋ x) → Lift (suc n) (x ∋ y) → Lift (suc n) (Union X ∋ y) |
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148 union→ X x y (lift X∋x) (lift x∋y) = lift lemma where |
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149 lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y |
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150 lemma {z} X∋z = {!!} |
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151 -- _∋_ {n} a x = def a ( od→ord x ) |
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152 ¬∅ : (x : OD {n} ) → ¬ x ≡ od∅ → Ordinal {n} |
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153 ¬∅ = {!!} |
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154 ¬∅∈ : (x : OD {n} ) → (not : ¬ x ≡ od∅ ) → x ∋ (ord→od (¬∅ x not)) |
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155 ¬∅∈ = {!!} |
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156 minimul : OD {n} → ( OD {n} ∧ OD {n} ) |
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157 minimul x = {!!} |
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158 regularity : (x : OD) → ¬ x ≡ od∅ → |
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159 Lift (suc n) (x ∋ proj1 (minimul x)) ∧ |
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160 (Select (proj1 (minimul x ) , x) (λ x₁ → Lift (suc n) (proj1 ( minimul x ) ∋ x₁) ∧ Lift (suc n) (x ∋ x₁)) ≡ od∅) |
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161 proj1 ( regularity x non ) = lift lemma where |
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162 lemma : def x (od→ord (proj1 (minimul x))) |
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163 lemma = {!!} |
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164 proj2 ( regularity x non ) = {!!} |